A strengthening of a theorem of Bourgain-Kontorovich-V

2016/04/17 by Kan, I. D.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1604.04884

Abstract

Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator (continuant) of a finite continued fraction (b)/(d)=[d₁,d₂,…,d_k], with all partial quotients d₁,d₂,…,d_k being bounded by an absolute constant A. Recently (in 2011) several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A=50 has positive proportion in ℕ. In 2014 Kan and Frolenkov proved this result with A=5. Let \mathfrakC_A be the set of infinite continued fractions whose partial quotients belong to A \mathfrakC_A=\[d₁,…,d_j,…]: d_j\inA, j=1,…\ and let δ be the Hausdorff dimension of \mathfrakC_A. Naw this result proved with A=4 and δ>0.7807….

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